TSTST 2025 Problem 2

Find all sets $S \subseteq \mathbb{Z}$ for which there exists a function $f\colon \mathbb{R} \to \mathbb{Z}$ such that

• $f(x-y) - 2f(x) + f(x+y) \ge -1$ for all $x, y \in \mathbb{R}$, and
• $S = \{f(z) \mid z \in \mathbb{R}\}$.